Cardinality

||A|| is the cardinality of the set A. Single vertical bars can also be placed around a set to mean cardinality, such as |A|.^[1][2]

The cardinality of a finite set is a natural number. The smallest cardinality is 0. The empty set has cardinality 0. If the cardinality of the set A is n, then there is a "next larger" set with cardinality n+1 (for example, the set A ∪ {A}). If ||A|| ≤ ||B|| ≤ ||A ∪ {A}||, then either ||B|| = ||A|| or ||B|| = ||A ∪ {A}||.) There is no largest finite cardinality.

If the cardinality of a set is not finite, then the cardinality is infinite.^[4]

An infinite set is considered countable if they can be listed without missing any (that is, if there is a one-to-one correspondence between it and the set of natural numbers ${\displaystyle \mathbb {N} }$ ).^[3] Examples include the rational numbers, integers, and natural numbers. Such sets have a cardinality that we call ${\displaystyle \aleph _{0}}$  (pronounced "aleph null", "aleph naught" or "aleph zero"). Sets such as the real numbers are not countable, since given any finite or infinite list of real numbers, it's always possible to find a number that's not on that list. The real numbers have a cardinality of ${\displaystyle {\mathfrak {c}}}$ —the cardinality of the continuum.^[1]

• Bijection
• Cardinal number
• Cantor's diagonal argument
• Cantor's theorem
• Continuum hypothesis
• Injective function